In this paper we study domains, Scott
domains, and the existence of measurements. We
use a space created by D.~K. Burke to show that
there is a Scott domain $P$ for which $\max(P)$ is
a $G_\delta$-subset of $P$ and yet no measurement
$\mu$ on $P$ has $\ker(\mu) = \max(P)$. We also
correct a mistake in the literature asserting that
$[0, \omega_1)$ is a space of this type. We show
that if $P$ is a Scott domain and $X \subseteq
\max(P)$ is a $G_\delta$-subset of $P$, then $X$
has a $G_\delta$-diagonal and is weakly
developable. We show that if $X \subseteq
\max(P)$ is a $G_\delta$-subset of $P$, where
$P$ is a domain but perhaps not a Scott domain,
then $X$ is domain-representable,
first-countable, and is the union of dense,
completely metrizable subspaces. We also
show that there is a domain $P$ such that
$\max(P)$ is the usual space of countable
ordinals and is a $G_\delta$-subset of $P$ in
the Scott topology. Finally we show that the
kernel of a measurement on a Scott domain can
consistently be a normal, separable,
non-metrizable Moore space.